Lecture 2 Describing and Visualizing Data

1. 2-1 Overview 2-2 Frequency Distributions 2-3 Visualizing Data Lecture 2Describing and Visualizing Data

2. Descriptive Statistics summarize or describethe important characteristics of a known set of data Inferential Statistics use sample data to make inferences (or generalizations) about a population 2.1Overview

3. Estimation & Inference • Method of concluding • Based on statistics f(X) Population Sample : Characteristics X: Sample values Sampling • Method of drawing • Based on probability Relationship

4. 1. Center: A representative or average value that indicates where the middle of the data set is located 2. Variation: A measure of the amount that the values vary among themselves 3. Distribution: The nature or shape of the distribution of data (such as bell-shaped, uniform, or skewed) 4. Outliers: Sample values that lie very far away from the vast majority of other sample values Important Characteristics of Data

5. Frequency Distribution lists data values (either individually or by groups of intervals), along with their corresponding frequencies or counts 2.2 Frequency Distributions

6. The Frequency of a ungrouped or regular data (discrete variable) - One value per category - how often it takes these value Gender Frequency Percent (Relative Frequency) 1 (Male) 20 40% 2 (Female) 30 60% Total 50 100% Frequency Distribution Tablefor ungrouped data

7. Example: Cotinine levels in smokers

8. Grouped Frequency Distribution Table

9. are the smallest numbers that can actually belong to different classes Lower Class Limits Lower Class Limits

10. are the largest numbers that can actually belong to different classes Upper Class Limits Upper Class Limits

11. number separating classes - 0.5 99.5 199.5 299.5 399.5 499.5 Class Boundaries

12. Class midpoints can be found by adding the lower class limit to the upper class limit and dividing the sum by two. Class Midpoints Class Midpoints 49.5 149.5 249.5 349.5 449.5

13. is the difference between two consecutive lower class limits or two consecutive lower class boundaries 100 100 100 100 100 Class Width Class Width

14. 1. Large data sets can be summarized. 2. Can gain some insight into the nature of data. Have a basis for constructing graphs. Disadvantage: information lost Reasons for Constructing Frequency Distributions

15. 1. Decide on the number of classes (should be between 5 and 20) , say C=5 for the example: cotinine levels in smokers (subjective decision) 2. Calculate (round up). W=(491-0)/5=98.2 round to 100. Constructing A Grouped Frequency Table (highest value) – (lowest value) class width (W)  number of classes 3. Starting point: Begin by choosing a lower limit of the first class, 0 4. Using the lower limit of the first class and class width, proceed to list the lower class limits: 0, 100, 200, 300, 400. 5. List the lower class limits in a vertical column and proceed to enter the upper class limits: 99, 199, 299, 399, 499. Go through the data set putting a frequency count in the appropriate class for each data value. Add a caption for the frequency distribution table

16. Relative Frequency Distribution class frequency relative frequency = sum of all frequencies 11/40 = 28% 12/40 = 40% etc. Total Frequency = 40

17. Cumulative Frequency Distribution Cumulative Frequencies

18. Frequency Tables

19. Recap • In this Section we have discussed • Important characteristics of data • Frequency distributions • Procedures for constructing frequency distributions • Relative frequency distributions • Cumulative frequency distributions

20. 2.3 Visualizing Data Depict the nature of shape or shape of the data distribution

21. Frequency Histogram Figure 2-1 A bar graph in which the horizontal scale represents the classes of data values and the vertical scale represents the frequencies.

22. Relative Frequency Histogram Figure 2-2 Has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies.

23. Histogram and Relative Frequency Histogram Figure 2-1 Figure 2-2

24. Frequency Polygon Figure 2-3 Uses line segments connected to points directly above class midpoint values

25. Ogive Figure 2-4 A line graph that depicts cumulative frequencies

26. Dot Plot Figure 2-5 Consists of a graph in which each data value is plotted as a point along a scale of values

27. Stem-and Leaf Plot Represents data by separating each value into two parts: the stem (such as the leftmost digit) and the leaf (such as the rightmost digit)

28. Pareto Chart Figure 2-6 A bar graph for qualitative data, with the bars arranged in order according to frequencies

29. Pie Chart Figure 2-7 A graph depicting qualitative data as slices of a pie

30. Scatter Diagram A plot of paired (x,y) data with a horizontal x-axis and a vertical y-axis

31. Time Series Plot Figure 2-8 Data that have been collected at different points in time

32. Other Graphs Figure 2-9 Innermost region: deaths from wounds Middle region: Deaths from other causes Outmost region: deaths due to preventable diseases

33. Recap In this Section we have discussed graphs that are pictures of distributions. Keep in mind that the object of this section is not just to construct graphs, but to learn something about the data sets – that is, to understand the nature of their distributions.